Perfect Binary Codes Research Articles

On the weight distribution of perfect binary codes

In this paper, we give a new proof of the closed-form formula for the weight distribution of a perfect binary single-error-correcting code.

On the Generalized Concatenated Construction for Codes in $${L_1}$$ and Lee Metrics

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2m we construct codes over $$\mathbb{Z}_4$$ with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2m+1 (with code distance 4). We construct codes over $$\mathbb{Z}_4$$ of length n with Lee distance n which under the Gray mapping yield Hadamard matrices of order 2n (under the additional condition that an Hadamard matrix of order n exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.

STOLARSKY'S INVARIANCE PRINCIPLE FOR FINITE METRIC SPACES

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.

Obtaining Binary Perfect Codes Out of Tilings

A tiling of $\mathbb {F}_{2}^{n}$ (considered as a graph) gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on $\mathbb {F}_{2}^{n}$ that are determined by a weight which respects the support of vectors (TS-metrics) and in this case, a perfect code is called TS-perfect. We consider all the tilings of $\mathbb {F}_{2}^{n}$ by tiles with up to 8 elements (which are classified in the literature) and determine which of them give rise to a TS-perfect code. In the sequence, for those tilings that give rise to a perfect code for some TS-metric, we classify the TS-metrics (up to equivalence) that turn it into a perfect code. Also, we give a general construction to obtain TS-perfect codes out of given TS-perfect codes, by proving that concatenation of two TS-perfect codes is by itself a TS-perfect code.

Perfect binary codes of infinite length

A subset C of infinite-dimensional binary cube is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.

On the symmetry group of the Mollard code

For a pair of given binary perfect codes C and D of lengths t and m respectively, the Mollard construction outputs a perfect code M(C,D) of length tm + t + m, having subcodes C1 and D2, that are obtained from codewords of C and D respectively by adding appropriate number of zeros. In this work we generalize of a result for symmetry groups of Vasilev codes [2] and find the group Stab_{D2}Sym(M(C,D)). The result is preceded by and partially based on a discussion of linearity of coordinate positions (points) in a nonlinear perfect code (non-projective Steiner triple system respectively).

On separability of the classes of homogeneous and transitive perfect binary codes

By the example of perfect binary codes, we prove the existence of binary homogeneous nontransitive codes. Thereby, taking into account previously obtained results, we establish a hierarchical picture of extents of linearity for binary codes; namely, there is a strict inclusion of the class of binary linear codes in the class of binary propelinear codes, which are strictly included in the class of binary transitive codes, which, in turn, are strictly included in the class of binary homogeneous codes. We derive a transitivity criterion for perfect binary codes of rank greater by one than the rank of the Hamming code of the same length.

Affine 3-nonsystematic perfect codes of length 15

A perfect binary code C of length n = 2k − 1 is called affine 3-systematic if some 3-dimensional subspace L exists in the space {0, 1}n such that the intersection of each of its cosets L + u with C is either empty or a singleton. Otherwise, the code C is called affine 3-nonsystematic. In the paper, we construct four nonequivalent affine 3-nonsystematic codes of length 15 and study the properties of the complement {0, 1}n \ (C + C).

Affine 3-nonsystematic codes

A perfect binary code C of length n = 2k − 1 is called affine 3-systematic if in {0, 1}n there exists a 3-dimensional subspace L such that the intersection of its every coset L + u with C is either empty or a singleton. Otherwise, C is called affine 3-nonsystematic. In this article we construct affine 3-nonsystematic codes of length n = 2k − 1 for k > 4.

Steiner quadruple systems of small rank and extended perfect binary codes

Using the switching method, we give a classification for the Steiner quadruple systems of orderN > 8 and rank r N (different by 2 from the rank of the Hamming code of length N) which are embedded into the extended perfect binary codes of length N and the same rank. Some lower and upper bounds are provided on the number of these different systems. The lower bound and description of different Steiner quadruple systems of order N and rank r N which are not embedded into the extended perfect binary codes of length N and the same rank are given.

On the steiner triple systems of small rank embedded into perfect binary codes

Using the switching method, some classification is pointed out of STS(n) of order n = 2r −1, r > 3, and small rank rn (different by 2 from the rank of the Hamming code of length n), embedded into perfect binary codes of length n and the same rank. The lower and upper bounds on the number of such different STSs are provided.Description of the class of STS(n)s of rank rn, which are not embedded into the perfect binary codes of length n and the same rank, and the lower bound on the number of these different systems are given. It is proved that each STS(n) of rank rn − 1 is embedded into a Vasil’ev code of length n and the same rank.

Structure of steiner triple systems S(2 m − 1, 3, 2) of rank 2 m − m + 2 over $\mathbb{F}_2 $

The structure of all different Steiner triple systems S(2 m −1, 3, 2) of rank 2 m −m+2 over $\mathbb{F}_2 $ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order 2 m − 1 and rank ≤ 2 m − m + 2. The number of such different systems of order 2 m − 1 and rank less than or equal to 2 m − m + 2 which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order 2 m − 1 and rank ≤ 2 m − m + 2 are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank.

On the Number of Nonequivalent Propelinear Extended Perfect Codes

The paper proves that there exists an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov (2007) are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation.

On the Steiner quadruple systems of small rank which are embeddable into extended perfect binary codes

The codewords of weight 4 of every extended perfect binary code that contains the all- zero vector are known to form a Steiner quadruple system. We propose a modification of the Lindner construction for the Steiner quadruple system of order N =2 r which can be described by special switchings from the Hamming Steiner quadruple system. We prove that each of these Steiner quadruple systems is embedded into some extended perfect binary code constructed by the method of switching of ijkl-components from the binary extended Hamming code. We give the lower bound for the number of different Steiner quadruple systems of order N with rank at most N − log N +1 which are embedded into extended perfect codes of length N.

A note on the symmetry group of full rank perfect binary codes

It is proved that the size of the symmetry group Sym(C) of every full rank perfect 1-error correcting binary code C of length n is less than or equal to 2|Sym(Hn)|/(n+1), where Hn is a Hamming code of the same length.

Structural properties of binary propelinear codes

The paper deals with some structural properties of propelinear binary codes, in particular propelinear perfect binary codes. We consider the connection of transitive codes with propelinear codes and show that there exists a binary code, the Best code of length 10, size 40 and minimum distance 4, which is transitive but not propelinear. We propose several constructions of propelinear codes and introduce a new large class of propelinear perfect binary codes, called normalized propelinear perfect codes. Finally, based on the different values for the rank and the dimension of the kernel, we give a lower bound on the number of nonequivalent propelinear perfect binary codes.

On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$

It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank$r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length$n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.

Involutions in Binary Perfect Codes

Given a 1-perfect code C, the group of symmetries of C, Sym(C)={π ∈ Sn | π(C)=C} , is a subgroup of the group of automorphisms of C. In this paper, we focus on symmetries of order two, i.e., involutions. Let InvF(C) ⊆ Sym(C) be the set of involutions that stabilize F pointwise. For linear 1-perfect codes, the possibilities for the number of fixed points |F| are given, establishing lower and upper bounds. For any m ≥ 2 and any value k between these bounds, [m/2] ≤ k ≤ m-1, linear 1-perfect codes of length n=2m-1 which have an involution that fixes |F| = 2k-1 coordinates are constructed. Moreover, for any m ≥ 4, 1 ≤ r ≤ m-1, and [m/2] ≤ k ≤ m-1, nonlinear 1-perfect codes of length n=2m-1 having rank n-m+r and an involution that fixes 2k-1 coordinates are also constructed, except one case, when m ≥ 6 is even, r=m-1 and k = [m/2].

On the structure of non-full-rank perfect $q$-ary codes

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.

On switching equivalence of n-ary quasigroups of order 4 and perfect binary codes

We prove that arbitrary n-ary quasigroups of order 4 can be transformed into each other by successive switchings of {a, b}-components. We prove that perfect (closely packed) binary codes with distance 3 whose rank (dimension of the linear span) is greater by 1 or 2 than the rank of a linear perfect code can be taken to each other by successive switchings of i-components.